Problem: Solve for $x$ and $y$ using elimination. $\begin{align*}-6x+3y &= -6 \\ 9x-9y &= 3\end{align*}$
We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $3$ and the bottom equation by $1$ $\begin{align*}-18x+9y &= -18\\ 9x-9y &= 3\end{align*}$ Add the top and bottom equations. $-9x = -15$ Divide both sides by $-9$ and reduce as necessary. $x = \dfrac{5}{3}$ Substitute $\dfrac{5}{3}$ for $x$ in the top equation. $-6( \dfrac{5}{3})+3y = -6$ $-10+3y = -6$ $3y = 4$ $y = \dfrac{4}{3}$ The solution is $\enspace x = \dfrac{5}{3}, \enspace y = \dfrac{4}{3}$.